Best Known (67, 84, s)-Nets in Base 16
(67, 84, 131088)-Net over F16 — Constructive and digital
Digital (67, 84, 131088)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (59, 76, 131071)-net over F16, using
- net defined by OOA [i] based on linear OOA(1676, 131071, F16, 17, 17) (dual of [(131071, 17), 2228131, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1676, 1048569, F16, 17) (dual of [1048569, 1048493, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1676, 1048569, F16, 17) (dual of [1048569, 1048493, 18]-code), using
- net defined by OOA [i] based on linear OOA(1676, 131071, F16, 17, 17) (dual of [(131071, 17), 2228131, 18]-NRT-code), using
- digital (0, 8, 17)-net over F16, using
(67, 84, 1048610)-Net over F16 — Digital
Digital (67, 84, 1048610)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1684, 1048610, F16, 17) (dual of [1048610, 1048526, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1683, 1048608, F16, 17) (dual of [1048608, 1048525, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1651, 1048576, F16, 11) (dual of [1048576, 1048525, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(167, 32, F16, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(1683, 1048609, F16, 16) (dual of [1048609, 1048526, 17]-code), using Gilbert–Varšamov bound and bm = 1683 > Vbs−1(k−1) = 682375 934352 272467 710614 302455 931600 613801 454964 372929 533539 633528 191006 338985 131069 790872 499421 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(1683, 1048608, F16, 17) (dual of [1048608, 1048525, 18]-code), using
- construction X with Varšamov bound [i] based on
(67, 84, large)-Net in Base 16 — Upper bound on s
There is no (67, 84, large)-net in base 16, because
- 15 times m-reduction [i] would yield (67, 69, large)-net in base 16, but